Torsion Type Invariants of Singularities

Abstract

Inspired by the LG/CY correspondence, we study the local index theory of the Schrödinger operator associated to a singularity defined on \(\mathbb {C}^{n}\) by a quasi-homogeneous polynomial f. Under some mild assumption to f, we show that the small time heat kernel expansion of the corresponding Schrödinger operator exists and is a series of fractional powers of time t. Then we prove a local index formula which expresses the Milnor number of f by a Gaussian type integration. The heat kernel expansion provides the spectral invariants of f. Furthermore, we can define the torsion type invariants associated to a homogeneous singularity. The spectral invariants provide another way to classify the singularity.